| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s + 13-s + 15-s − 19-s + 3·21-s + 6·23-s + 25-s − 27-s + 6·29-s + 5·31-s + 3·35-s − 5·37-s − 39-s + 8·41-s + 8·43-s − 45-s + 4·47-s + 2·49-s − 12·53-s + 57-s + 12·59-s + 3·61-s − 3·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.229·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.898·31-s + 0.507·35-s − 0.821·37-s − 0.160·39-s + 1.24·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s + 2/7·49-s − 1.64·53-s + 0.132·57-s + 1.56·59-s + 0.384·61-s − 0.377·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 377520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.965442766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965442766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.52734355153555, −12.09537768431930, −11.45761866591298, −11.26574888007711, −10.67850506237503, −10.16220993360322, −9.987791648915069, −9.252244813736998, −8.898670115720695, −8.484828214556876, −7.842640985326348, −7.283847839893164, −6.959697894458835, −6.439205568137382, −6.054187821357309, −5.616763244013171, −4.913167407505169, −4.491538138403590, −4.040955157015399, −3.347452032381304, −2.971488290829202, −2.462999125986658, −1.591755740547124, −0.7905232495357133, −0.5289622180479596,
0.5289622180479596, 0.7905232495357133, 1.591755740547124, 2.462999125986658, 2.971488290829202, 3.347452032381304, 4.040955157015399, 4.491538138403590, 4.913167407505169, 5.616763244013171, 6.054187821357309, 6.439205568137382, 6.959697894458835, 7.283847839893164, 7.842640985326348, 8.484828214556876, 8.898670115720695, 9.252244813736998, 9.987791648915069, 10.16220993360322, 10.67850506237503, 11.26574888007711, 11.45761866591298, 12.09537768431930, 12.52734355153555
